一个改进的影响球图大小的上界
An improved upper bound for the size of the sphere of influence graph
摘要：
设$V$是平面中的一组$n$点。对于每个$x\in V$，令$B_x$为以$x$为中心的闭合圆盘，其半径等于从$x$到其最近邻居的距离。定义$V$上的{ \it闭影响球图}为$x$与$y$相邻的无向图当且仅当$B_x$与$B_y$有非空交集.已知对于某个绝对正的常数$c$，每个$n$ -顶点闭影响球图至多有$cn$条边.第一个结果是由Avis和Horton在1985年获得的，他们提供了值$c=29$。他们的结果被几位作者相继改进：Bateman和Erdalys（c=18），Michael和Quint（c=17.5），以及Soss（c=15）。在本文中，我们证明了一个可以采取$c=14.5$。
英文摘要：
Let $V$ be a set of $n$ points in the plane. For each $x\in V$, let $B_x$ be the closed circular disk centered at $x$ with radius equal to the distance from $x$ to its closest neighbor. The {\it closed sphere of influence graph} on $V$ is defined as the undirected graph where $x$ and $y$ are adjacent if and only if the $B_x$ and $B_y$ have nonempty intersection. It is known that every $n$-vertex closed sphere of influence graph has at most $cn$ edges, for some absolute positive constant $c$. The first result was obtained in 1985 by Avis and Horton who provided the value $c=29$. Their result was successively improved by several authors: Bateman and Erdős (c=18), Michael and Quint (c=17.5), and Soss (c=15). In this paper we prove that one can take $c=14.5$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
